Garry O'Connor led the way with a header before Rangers' nemesis Ivan Sproule added a second.
When substitute Chris Killen scored a third for the visitors 12 minutes from the end, that was the signal for the home fans to head for the exit.
In the end it was a deserved victory for Hibs, at half-time Rangers looked the more likely winners.
Simon Brown in the Hibs goal did well to cut out a Chris Burke cross as Rangers players queued at the back post to apply the finishing touch.
The home side squandered two good chances when Kris Boyd and then Peter Lovenkrands glanced headers wide.
Brown then surpassed himself with a marvellous fingertip save from Robert Malcolm's fierce shot which the keeper touched on to the bar.
Hibs' only threat in the first-half came from Sproule and it was no surprise when the pacy striker set up their first goal. His cross to the back post was powerfully headed home by O'Connor.
Sproule added a second when he drove his shot low past Ronald Waterreus.
Rangers looked to have pulled a goal back when Sotirios Kyrgiakos headed home, but the referee spotted a foul by Boyd, who was also booked.
Hibs finished the tie as a contest when Derek Riordan's inch-perfect pass found Killen, whose first shot was blocked by Waterreus before he sidefooted the rebound home from an acute angle.